This article was downloaded by: [University of Connecticut]On: 11 October 2014, At: 05:45Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Structure and Infrastructure Engineering:Maintenance, Management, Life-Cycle Design andPerformancePublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/nsie20

Comparison of algorithms for seismic topologyoptimisation of lifeline networksWei Liuab & Jie Liab

a Department of Building Engineering, Tongji University, Shanghai 200092, P.R. Chinab State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University,Shanghai 200092, P.R. ChinaPublished online: 23 Jul 2013.

To cite this article: Wei Liu & Jie Li (2014) Comparison of algorithms for seismic topology optimisation of lifeline networks,Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance, 10:11, 1357-1368,DOI: 10.1080/15732479.2013.808234

To link to this article: http://dx.doi.org/10.1080/15732479.2013.808234

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Comparison of algorithms for seismic topology optimisation of lifeline networks

Wei Liua,b* and Jie Lia,b

aDepartment of Building Engineering, Tongji University, Shanghai 200092, P.R. China; bState Key Laboratory for Disaster Reduction inCivil Engineering, Tongji University, Shanghai 200092, P.R. China

(Received 3 June 2012; final version received 8 April 2013; accepted 17 May 2013; published online 23 July 2013)

A performance comparison among five optimisation algorithms for the topology design of lifeline network subjected toearthquakes is presented in this study. The topology optimisation model in conjunction with the argument of seismicreliability is firstly introduced for the design of lifeline networks subjected to earthquakes. Subsequently, two quite standardoptimisers used in the numerical optimisation, i.e. a genetic algorithm (GA) and a simulated annealing algorithm, areinvestigated. Their hybrid scheme, entitled a simulated annealing GA that combines the advantages of two standardoptimisers, is introduced as well. Besides, an ant colony algorithm and a particle swarm algorithm that have been developedin recent years are explored. Three modelled lifeline networks, including two benchmark networks and one actual network,are used as the numerical carriers that the five optimisation algorithms accommodate. It is concluded that the simulatedannealing GA provides an excellent tool with higher efficiency to achieve optimal topology of lifeline networks.

Keywords: lifeline networks; seismic design; genetic algorithm; simulated annealing algorithm; hybrid optimisers; antcolony algorithm

1. Introduction

Lifeline systems, such as water distribution, gas supply and

power networks, are essential to modern cities (Li, 2005).

Damages of these systems subjected to strong earthquakes

not only cause economic losses, but also disrupt residential,

commercial and industrial activities. For example, in the

1995 Kobe earthquake the water distribution network

suffered extensive damage (Investigation Group of Kobe

Earthquake, 1997). The number of leaks and breaks was as

large as 1610. As a result, about 1,360,000 customers

encounteredwater supply cut and an additional disaster was

caused due to fire which led to even more serious damages,

since many fire hydrants could not supply water.

Great research interest has been stimulated in seismic

analysis of lifeline systems, including seismic analysis of

network components (Muleski & Ariman, 1985; Takada

et al., 2000), seismic reliability analysis of networks

(Hwang, Lin, & Shinozuka, 1998; Li & He, 2002) and

seismic interaction between different lifeline systems

(Duenas-Osorio, Craig, & Goodno, 2007). However, it is

not the seismic analysis but the analysis-based design and

retrofit being the ultimate goal of the research in this field.

For this purpose, Shinozuka, Tan, and Koibe (1981)

investigated the seismic optimisation of a simple water

distribution network. Based on the seismic connectivity

reliability analysis of lifeline networks, Bao (2004) and Liu

and Li (2008) presented a simulated annealing algorithm

(SAA) to find a least-cost network, of which the seismic

reliability satisfies a prescribed value. Furthermore, Li, Liu,

and Bao (2008) proposed a topology optimisation model

based on the seismic connectivity reliability analysis of

lifeline networks and adopted a genetic algorithm (GA) to

solve this optimisation problem. Along the same lines, a

simulated annealing GA was implemented to solve this

problem (Li & Liu, 2008, 2009). For a water distribution

network, seismic functional reliability analysis can better

describe its state after an earthquake. Hence, Liu, Xu, and

Li (2012a, 2012b) proposed an optimisation methodology

to design water distribution networks subjected to earth-

quakes. For this purpose, several optimisation algorithms,

including a GA, a simulated annealing GA, an ant colony

algorithm (ACA) and a particle swarm algorithm (PSA),

have been used and compared.

In this study, the five aforementioned optimisation

algorithms are compared in terms of their performance for

network topology optimisation of lifeline networks based

on seismic connectivity reliability analysis, as it is suitable

for any type of lifeline networks. First, a topology

optimisation model for the lifeline networks subjected to

earthquake is introduced by taking the system cost as the

optimisation objective and the system seismic reliability as

the constraint. Second, three optimisation algorithms, a GA

(Li et al., 2008), an SAA (Bao, 2004; Liu & Li, 2008) and a

simulated annealing genetic algorithm (SAGA; Li & Liu,

q 2013 Taylor & Francis

*Corresponding author. Email: liuw@mail.tongji.edu.cn

Structure and Infrastructure Engineering, 2014

Vol. 10, No. 11, 1357–1368, http://dx.doi.org/10.1080/15732479.2013.808234

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2008, 2009), a hybrid algorithm that combines the

advantages of GA and SAA are introduced to solve the

problem. In addition, two recently developed optimisation

algorithms, an ACA and a PSA, are introduced for the

problem. In order to compare these five optimisation

algorithms, two benchmark networks and an actual network

are used. The results demonstrate that SAGA has the best

performance, GA performs well, SAA performs moder-

ately, ACA follows and PSA performs rather poorly for the

seismic topology optimisation of lifeline networks.

2. Topology optimisation model

A lifeline network is usually modelled by a graph G ¼ (N,

E), where N is a set of nodes and E is a set of edges. Take a

gas supply network as an example, where the nodes may

represent the consumers or the gas factories, and the edges

are the pipelines between pairs of nodes. Moreover, the gas

factories are defined as the sources while the other nodes

are defined as the terminals. To fulfil an optimisation

design is actually to find an optimal network topology with

least cost while the seismic reliabilities of its nodes satisfy

a prescribed threshold reliability. An optimisation model

can be mathematically formulated in the following general

form (Li et al., 2008):

minimise CðG *Þ ¼ Pgij�cij

subject to Pk $ P0 k ¼ 1; 2; . . . ; n;ð1Þ

where G*, a subgraph of G, is an optimal solution; G

denotes a graph which is usually generated empirically

according to the geographic location of the network nodes

and roads; gij ¼ 0 if no edge exists between node i and

node j and otherwise gij ¼ 1; cij is the cost of the edge

between node i and node j; Pk is the seismic reliability

between terminal k and the source which can be obtained

by a recursive decomposition algorithm (RDA; Li & He,

2002); P0 is the prescribed threshold reliability and n is the

number of the terminals.

Apparently, solving the model can give the topology of

the lifeline networks. The optimal result describes the

layout of the network. Therefore, such an optimisation

process can be used in the planning phase of lifeline

networks design. For example, when a gas supply network

is optimised, the locations, materials and diameters of the

pipelines are clear. After that, the detailed design of the

network, such as pipelines connections, valves and

regulator stations, can be performed.

It has to be noted that the described approach can be

used to design a new network or to retrofit an old network.

It is noticed that G* is a subgraph of G. When designing a

new network, G is a scheme design with a certain

redundant edges. After optimisation, some edges are

removed from G and a planning network G* is generated

and proposed. Herein, all edges in G* have to be

constructed. When retrofitting an old network, G is a

network which adds many edges to the old network.

Similarly, after optimisation, an optimal network G* is

generated and proposed. Different from the new network,

some new edges, which are in the optimal network G* but

not in the old network, have to be constructed. In the

meantime, some old edges, which are in the old network

but not in the optimal network G*, can be discarded.

The optimisation goal of the model only considers the

edge’s cost. In fact, the costs of lifeline networks may

include many other elements. However, the costs of these

elements usually vary slightly. On the other hand, the

major influential factors for the seismic performance of

some lifeline networks are the edges while other elements

have little influence. Therefore, only taking the edges’ cost

as the optimisation objective is feasible. Take the gas

supply network as an example, in addition to the costs of

pipelines, the cost of network may include the cost of gas

factories. When the capacities of the factories are

determined, their costs are usually constants. In addition,

it was found that usually the pipelines suffered serious

damages while the damages of factories were usually

lighter during many previous earthquakes, such as the

Wenchuan Earthquake in 2008 (State Key Laboratory of

Disaster Reduction in Civil Engineering, 2008).

The problem above is a typical combinatorial

optimisation one. For a network G consisting of m edges

and n nodes, the number of all potential networks usually

equalsPm

i¼n21Cim $ 20:3nþ1 , 21:5nþ1 (Li et al., 2008).

For a large lifeline network that consists of more than 200

nodes, the number of all potential networks is an

astronomical figure and the time for calculating all

potential networks is unacceptable in engineering practice.

Nevertheless, the developed modern combinational

optimisation technology provides an effective tool for

the problem.

3. Optimisation algorithms for seismic optimisation

of lifeline networks

Before applying the optimisation algorithms on the model,

the network should be properly represented. Note that any

solution in the optimisation model is a subgraph of the

original network G. The simplest 0–1 binary coding can

be adopted here. An S bits array is used to represent a

graph and each bit represents an edge of the graph G,

where S is the number of edges in G. A ‘1’ in the array

means that the network possesses a corresponding edge of

G while a ‘0’ means not. For example, Figure 1 shows a

bridge network. In this figure, the original G includes all

the five edges while the subgraph does not include edge 5

(dash line). Therefore, the corresponding array of this

subgraph can be written as 11110.

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3.1 Genetic algorithms

GA was introduced by Holland during the mid-1970s in

the field of machine learning (Holland, 1975). It is a global

search method which imitates the natural selection process

(Holland, 1992), and has been used in various applications

in civil engineering, such as bridge maintenance

optimisation at the transportation network level (Bocchini

& Frangopol, 2011; Liu & Frangopol, 2006). When GA is

used for the seismic reliability optimisation of lifeline

networks, a generation including many genes is initially

created with each gene representing a network. Then by

using the selection, crossover and mutation operators, a

new generation is evolved. After a number of iterations, a

near-global optimal solution can be found in many cases.

The process of GA can be described as follows (Li

et al., 2008):

(1) Produce a group of solutions randomly as the

initial generation.

(2) Calculate the genes’ fitnesses in the generation.

(3) Use the selection, crossover and mutation

operations on the genes of current generation and

produce a new generation.

(4) Check whether the number of the iterations has

reached a prescribed value, if yes, stop and output

the result, otherwise return to step 2.

3.2 Simulated annealing algorithm

SAA was introduced by Kirkpatrick, Gelatt, and Vecchi

(1983) and also independently by Cerny (1985) as a

problem-independent combinatorial optimisation tech-

nique. In SAA, a network is seen as a current solution of

the model. Then, it is continually compared with the

possible solution ‘close’ to it, which is given by perturbing

the current solution, and updated at a probability described

as follows:

Pði ) jÞ ¼1 f ðjÞ # f ðiÞ;

exp f ðiÞ2f ðjÞt

� �f ðjÞ . f ðiÞ;

8<: ð2Þ

where t is the current temperature, a control parameter

which gradually decreases and approaches 0 at the end of

the process, while solution i and j are the current solution

and the new solution after perturbation, respectively; f ðiÞ isthe energy function of network i that can be defined as:

f ðiÞ ¼CðiÞ Pmin i $ P0;

Mþa P0 2Pmin i½ � þbPsumi Pmin i , P0

(; ð3Þ

where CðiÞ is the cost of the network i; Pmin i is the

minimum nodal reliability of network i; M is a large

constant and usually takes the value of the cost of network

G; Psumi ¼P

j;P0.PijðP0 2 PijÞ, Pij is the jth nodal seismic

reliability of network i; a and b are constants and their

values are determined by the actual networks data.

Apparently, if the perturbation result is an improved

solution, it is accepted and the current solution is updated

accordingly. Otherwise, it can also be accepted at a certain

probability. By accepting a worse solution, SAA avoids

being trapped too early into a local optimal solution

region. In addition, the probability of accepting a worse

perturbation solution decreases as t decreases as the

process of SAA advances, which guarantees that the

algorithm can eventually converge and will be less likely

to move away from a global optimal solution after having

approached it.

For the network topology optimisation, the process of

SAA can be described as follows (Bao, 2004; Liu & Li,

2008):

(1) Produce an initial network randomly as the current

solution.

(2) Determine the current temperature, a function of

the initial temperature and the cooling schedule. If

the current temperature is lower than a prescribed

small positive number, then stop.

(3) Perturb the current solution and generate a new

solution. Calculate the new solution’s energy

function and determine its acceptance probability.

(4) Generate a number varying between 0 and 1 at

random and compare it with the acceptance

probability. If the random number is smaller than

the acceptance probability, the new solution is

accepted and the current solution is updated.

Otherwise, the new solution is discarded and the

current solution is preserved.

(5) Check whether the number of perturbations has

reached the prescribed value at the current tempera-

ture, if yes, go to step 2, otherwise go to step 3.

3.3 A hybrid algorithm-simulated annealing GA

Although GA is a very powerful tool for the combinatorial

optimisation problem, it has a deficiency, i.e. premature

convergence, which occurs when GA gets trapped into a

local optimum and cannot escape from it. Strengthening

the searching ability is a good way to overcome the

1 4

5

2s t3

Figure 1. A typical bridge network.

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deficiency. As SAA is capable of widening the searching

space and its operator is very similar to the mutation

operator in GA, by replacing the mutation operator in GA

with the perturbation in SAA, a hybrid algorithm, the so-

called SAGA, can be developed to strengthen the

searching ability of GA (Li & Liu, 2008, 2009; Liu

et al., 2012a). Therefore, each gene generated by the

crossover operator will be perturbed to produce several

new solutions and updated according to the acceptance

probability in Equation (2).

3.3.1 Generating initial graphs

To generate initial graphs, a gene is initialised to contain

no edges, i.e. the bits in the array of the gene are all ‘0’s’.

Consequently, each bit in the array is chosen in turn and

changed to ‘1’ at a prescribed probability. In other words,

each edge in G is randomly added to the graph. The

process is repeated several times and the initial generation

is produced.

3.3.2 Evaluating graphs

For each generation, all genes need to be evaluated to

determine their fitnesses, an evaluation function of gene.

The fitness of gene k can be written as:

TðkÞ ¼ R2 CðkÞ2 QðkÞ; ð4Þ

where R is a large constant, TðkÞ is the fitness of gene k, C(k) denotes the cost of gene k and QðkÞ is a penalty factor.

Herein, QðkÞ is introduced because some genes, which do

not satisfy the prescribed threshold reliability, are not

feasible solutions. Herein, the penalty factor of gene k can

be defined as:

QðkÞ ¼0 Pmink $ P0;

max ðPmin jÞ2Pmink

� �Cmax2Cmin

max ðPmin jÞ2min ðPmin jÞ Pmink , P0

8<: ;

ð5Þ

where Pmin j denotes the minimal nodal seismic reliability

of gene j and can be obtained by comparing all nodal

reliabilities of gene j; max ðPmin jÞ and min ðPmin jÞ,respectively, represent the maximum and minimum values

of Pmin j among all genes in a generation; Cmax and Cmin

denote the maximum and minimum gene cost in current

generation, respectively.

3.3.3 Selection and crossover operators

The selection operator is a core operator, as it chooses the

superior genes, i.e. individuals with high fitness, at a high

probability and the inferior genes, individuals with low

fitness, at a low probability and passes them to the next

generation. Herein, a roulette wheel selection operator and

an elitist selection operator (Chen et al., 1996) are adopted,

i.e. the best network is preserved and one-third networks

are selected randomly based on their fitnesses and

preserved in the next generation. The detail process is

the same as that described in the work by Li et al. (2008).

The crossover operator is also an important one, which

guarantees the global searching capability of the

algorithm. This operator takes two genes at a prescribed

probability and produces two offsprings. In this study, a

one-point crossover operator (Chen et al., 1996) is

adopted. Just as its name implies, a random position in

both arrays is selected as the crossover point. After

exchanging the bits behind this position of two arrays, two

new networks are produced. In general, the process is the

same as that described in the work by Li et al. (2008).

3.3.4 Simulated annealing operator

In SAGA, a simulated annealing operator (SAO) replaces

the mutation operator to act on each gene produced by the

crossover operator. The process of this operator is to

perturb the gene to produce a new gene and update the old

gene at a probability determined by Equation (2). The

process is repeated several times until the number of the

perturbations reaches a prescribed value. The process of

perturbation is very simple and can be stated as follows:

(1) For each bit of the gene that should be perturbed,

produce a random number between [0, 1] and

compare it with a perturbation probability given

via Equations (6) and (7), from which the

perturbation probability of a bit changing from 0

to 1 can be determined by:

P01ðjÞ ¼ Pmax 01 þ Pmax 01 2 Pmin 01

I max 2 I min

ðIðjÞ2 I max Þ;ð6Þ

where Pmax 01 and Pmin 01 represent the maximum

and minimum probability of a bit changing from 0

to 1, respectively; I( j) represents the element

investment importance of element j, I max and Imin ,

respectively, represent themaximum andminimum

element investment importance in the network G.

The introduction of the element investment

importance is presented in the work by Li et al.

(2008). Similarly, the perturbation probability of a

bit changing from 1 to 0 can be determined by:

P10ðjÞ ¼ Pmin 10 þ Pmin 10 2 Pmax 10

I max 2 I min

ðIðjÞ2 I max Þ;ð7Þ

where Pmax 10 and Pmin 10 represent the maximum

and minimum probability of a bit changing from 1

to 0, respectively.

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(2) If the generated random number is smaller than the

perturbation probability, the bit is reversed which

means the bit is modified to 0 if it is 1, or 1 if it is 0.

(3) Check the connectivity of the new gene. If it is a

connected graph, the perturbation ends and a new

gene is produced. Otherwise, several edges are

randomly added to it. If it is still a disconnected

graph aftermodifying, it is discarded and go back to

step 1 to produce a new gene to replace it

Based on the above mentioned process, the process of

the SAO can be described as follows:

(1) Determine the current temperature, a function of

initial temperature and a cooling schedule which

can be expressed as:

t ¼ g l21T; ð8Þ

where T is the initial temperature which is a large

value determined by the actual network and l is the

number of current iteration, 0 , g , 1.

(2) Perturb the current gene to generate a new gene. Its

energy function is calculated using Equation (3)

and its acceptance probability is determined using

Equation (2).

(3) Generate a number varying between 0 and 1 at

random and compare it with the acceptance

probability of the new gene. If the random number

is smaller than the acceptance probability, then the

new gene is accepted and the current gene is

updated. Otherwise, the new gene is discarded and

the current gene is preserved.

(4) Check whether the number of perturbations has

reached a prescribed value, if yes, stop, otherwise

go to step 2.

Finally, the process of SAGA can be briefly described

as follows:

(1) Produce a group of networks randomly as the initial

generation.

(2) Calculate the genes’ fitnesses in the generation.

(3) Apply the selection operator, crossover operator

and SAO on the genes of current generation and

produce a new generation.

(4) Check whether the number of the iterations has

reached a prescribed value, if yes, stop and output the

result, otherwise return to step 2.

3.4 Ant colony algorithm

In 1996, Dorigo, Maniezzo, and Colorni (1996) proposed

an ACA by imitating the process that the ant colony looks

for food. When it is applied on the seismic topology

optimisation of the lifeline networks, some major steps can

be described as follows.

3.4.1 Pheromones initialisation

When an ACA is used, the pheromones, which determine

the selection probability of edges, should be initialised.

The network G owns S edges and each edge owns two

values, 0 and 1. Herein, 2S values are set as the initial

values of the pheromones, and ti;jð1Þ represents the initialpheromone of j ( j ¼ 0, 1) for bit i.

3.4.2 Generating networks

When generating a network, an ant is assumed to crawl

along each bit and to determine its value randomly. When

an ant crawls all bits in an array, a network is generated.

When an ant colony consists of m ants, m networks are

produced. During the above process, the probability that

ant k gives a value of j for bit i is

pki;jðrÞ ¼ti;jðrÞ

ti;0ðrÞ þ ti;1ðrÞ ; j ¼ 0; 1; ð9Þ

where ti;jðrÞ is the pheromone of j for bit i in the rth

iteration.

3.4.3 Network evaluation and pheromones updation

When the networks are generated, Equation (3) is used to

evaluate the generated networks. After evaluation, the best

network, which owns minimal energy function, is obtained

and compared with the best network of the previous

process. If the former is better, the latter is replaced by the

former. Then, the pheromones are updated. Herein, the

update model is

ti;jðr þ 1Þ ¼ ð12 rÞ�ti;jðrÞ þXmk¼1

Dtki;j; ð10Þ

Dtki;j ¼Qf ðkÞ ant k selects jwhen crawling bit i;

0 other

(; ð11Þ

where r is the pheromone trails evaporation coefficient,

0 # r # 1, m is the number of ants, Dtki;j is the pheromone

increment of bit i for value j corresponding to ant k, Q is

the total pheromone and usually a positive constant and

f ðkÞ is the energy function in Equation (3).

Therefore, the steps of the ACA can be described as

follows:

(1) Initialise all pheromones.

(2) Generate m networks using m ants and calculate

the energy functions of these networks using

Equation (3).

(3) Update the pheromones using Equations (10) and

(11).

(4) Check if the iterations reach a prescribed value. If

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yes, the algorithm ends and gives the best network

among the whole process. If not, return to step 2.

3.5 Particle swarm algorithm

By simulating the movements of bird flocks searching for

food, in 1995, Kennedy and Eberhart proposed a new

optimisation algorithm, the PSA, for combinational

optimisation problem. Afterwards, it has been widely

applied into various optimisation problems, such as

reactive power and voltage control (Yoshida, Kawata,

Fukuyama, Takayama, & Nakanishi, 1999) and finite

impulse response digital filters design (Li, Zhang, & Zhao,

2005).

3.5.1 Generating initial networks and initial velocity

arrays

Herein, the process of generating the initial group of

networks is as same as that of SAGA. Meanwhile, for each

network of the initial group, a corresponding velocity

array, which represents the variation rate of the bits in the

network, should be established. In this study, each bit in

the velocity array is given a randomly selected value

between 21 and 1.

3.5.2 Network evaluation and evolution

Equation (3) is used to evaluate the networks. In the

meantime, the best network Gc among the current group,

which owns minimal energy function, is determined. Then,

it is compared with the best solution of the whole process,

Gg. If it is better, Gg is updated by Gc. For each network

among the current group, using the following equations,

the velocity arrays are updated and the networks in group r

evolve to a new group of networks.

vrþ1ij ¼ v�vrij þ c1�rand1ðÞ�ðpric 2 xrijÞ

þ c2�rand2ðÞ�ðprig 2 xrijÞ; ð12Þ

xrþ1ij ¼ xrij þ vrþ1

ij ; ð13Þ

where vrij is the value of bit i in velocity array j; xrij is the

value of bit i in network j and v is a weight. When it is

large, the global searching performance of the algorithm

increases. In contrast, when it is small, the local searching

performance of the algorithm increases. In addition, pric is

the value of bit i in Gc, prig is the value of bit i in Gg, c1 and

c2 are acceleration constants and take values between 0

and 2. Finally, rand1ðÞ and rand2ðÞ are random functions

which generate the random numbers between 0 and 1.

Apparently, the value given by Equation (13) is not an

integer. As the bit of the network owns only two values, 0

and 1, it takes the value of 0 when it is smaller than 0.5 and

1 for the rest. Subsequently, in order to enhance the search

capability of the algorithm, each bit in the network is

reversed at a small described probability, i.e. it becomes 1

if it is 0 and 0 if it is 1. In this study, the probability takes

the value of 0.1.

Hence, the steps of the PSA can be stated as follows:

(1) Generate a group of networks as initial

group. Correspondingly, for each network, a

velocity array is established.

(2) Use Equation (3) to evaluate each network and

determine Gc and Gg.

(3) Update the velocity arrays and evolve the

networks to a new group of networks using

Equations (12) and (13).

(4) Check if the iterations reach a prescribed value. If

yes, the algorithm ends and gives Gg. If not, return

to step 2.

4. Case studies

The cost of the pipeline varies for different cities or areas.

In this study, the pipeline’s cost is evaluated by the

following equation (Wang, 2005):

c ¼ ð2144:36þ 4313:3dÞ�l; ð14Þ

where c is the pipeline’s cost in the unit of Yuan (RMB)

and d(m) and l(m) are the diameter and length of pipeline,

respectively.

4.1 Example 1

Figure 2 depicts a simple gas network, used by Li et al.

(2008), which has 10 nodes and 14 pipelines. Node 1 (the

bigger node) is the source and the seismic reliability of

each pipeline is assumed to be 0.9. The lengths and

diameters of pipelines are given in Table 1, and the total

cost is ¥59,197,800. For this network, the number of all

feasible topologies isP14

i¼9Ci14 ¼ 3473. All these topol-

ogies are analysed and the least-cost solutions that,

respectively, satisfy the threshold reliability of 0.7, 0.8 and

0.9 are shown in Figures 3–5. The total costs and the

10 14 13

109 11 12

4 6 7 8

3 4

2 1 1

5

5 6

9 8

7

3 2

Figure 2. A simple network.

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lowest nodal reliabilities of these optimal solutions are

given in Table 2. The results show that the total cost will

be higher with the increase in the threshold reliability.

The five optimisers, GA, SAA, SAGA, ACA and PSA,

are used to optimise the original network in Figure 2. The

basic parameters of the algorithms are as follows:

(1) GA: the number of the genes in a generation takes

the value of 90 and the number of generations is

50. The crossover probability is 0.8. Pmax 01,

Pmin 01, Pmax 10 and Pmin 10 take the values of 0.9,

0.6, 0.4 and 0.1, respectively.

(2) SAA: the number of the perturbations at each

temperature is set as 40. Pmax 01, Pmin 01, Pmax 10

and Pmin 10 take the values of 0.9, 0.6, 0.6 and 0.3,

respectively. Also, the initial temperature T takes

the value of 60,000, g takes the value 0.7 and the

terminal temperature tf takes the value of 0.001.

(3) SAGA: Pmax 01, Pmin 01, Pmax 10 and Pmin 10 take

the values of 0.9, 0.6, 0.6 and 0.3, respectively, and

the number of the perturbations is set as 5. The

other parameters are the same as those of GA or

SAA.

(4) ACA: the iteration number is 225 and the number

of ants takes the value of 20. All the initial

pheromones of selecting 0,ti;0ð1Þ, take the value of5 and those of selecting 1,ti;1ð1Þ, take the value of10. Q ¼ 500 and r ¼ 0:2.

(5) PSA: the number of networks in a group is 30 and

the number of iterations is 150. The weight of ith

generation is vi ¼ 1:52 1:1ði2 1Þ=149. Two

acceleration constants of ith generation are

c1i or c2i ¼ 2:02 ði2 1Þ=149.For each threshold reliability (0.7, 0.8 and 0.9), each

algorithm is repeated 1000 times and 1000 solutions are

obtained. The searching rates of the five algorithms, i.e.

the rate of obtaining the optimal solutions in Figures 3–5

among all runs, are given in Table 3. Apparently, the

performance of PSA is the weakest and its searching rate is

less than 10%. For the threshold reliability of 0.8, its

searching rate is even 3.9%. ACA performs a little better,

since although the searching rate of ACA for the threshold

reliability of 0.7 is only 1.5%, the rate for that of 0.8 and

0.9 reaches 14.5% and 21.2%. SAA performs better than

Table 1. Pipeline parameters of the network in Figure 3.

No. Length (m) Diameter (mm) No. Length (m) Diameter (mm)

1 1000 300 8 1000 4002 6500 250 9 4800 2003 4000 200 10 4000 2004 4000 300 11 4000 2505 4000 250 12 5000 3006 6500 250 13 6000 2007 6000 300 14 8500 200

4

3

5

3 2

10 89

6

9 10

4

11

1 1

5

6 8 7

13

Figure 3. The best solution of simple network when p0 is 0.7.

10 9 81314

9 11

4 5 6 8 7

3 4 5

113 2 2

Figure 4. The best solution of simple network when p0 is 0.8.

10 9 81314

9 11

4 5 6 8 7

3 4

10

5

113 2 2

Figure 5. The best solution of simple network when p0 is 0.9.

Table 2. The best solutions for the network in Figure 3.

Reliabilityconstraint ( p0)

Lowest nodalreliability Cost (RMB)

0.7 0.747 34,370,0000.8 0.849 37,610,0000.9 0.920 40,480,000

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ACA, its searching rate is between 20% and 30%. In

contrast, the performance of GA is good and its searching

rate ranges from 58% to 80%. Moreover, the performance

of SAGA, which can find the optimal solutions almost

each time, is perfect. The average searching rate of SAGA

is as high as 98.7%.

4.2 Example 2

Figure 6 depicts a medium gas network with 16 nodes and

24 edges, in which node 1 (the bigger node) is the source.

The pipelines’ reliabilities are randomly assigned and

presented in Table 4. The lengths and diameters of

pipelines are also given in Table 4. The total cost of this

network is ¥30,479,600. For this network, the number of

all feasible topologies isP24

i¼15Ci24 ¼ 2; 579; 130.

Obviously, the solution space is much larger than in

Example 1. In order to compare the performance of the

five algorithms, all these topologies are calculated and the

least-cost solutions that satisfy the threshold reliability of

0.7, 0.8 and 0.9 are shown in Figures 7–9, respectively.

The total costs and the lowest nodal reliabilities of these

solutions are given in Table 5.

All parameters of the five algorithms are the same as

those in Example 1. Each algorithm is repeated 100 times

for each threshold reliability as the computation time of

this network is much longer than that of Example 1.

Among these solutions, the searching rates of the five

algorithms are summarised in Table 6. Obviously, the

performance of SAGA is the best, then GA and SAA while

the performance of ACA and PSA is the worst. For this

Table 3. Searching rate for the network in Figure 3.

Algorithm P0 ¼ 0.7 (%) P0 ¼ 0.8 (%) P0 ¼ 0.9 (%)

GA 62.2 58.0 80.8SAA 32.6 20.1 20.8SAGA 97.9 98.7 99.5ACA 1.5 14.5 21.2PSA 6.3 3.9 9.0

1

4 5 6 7

11 12 13 14

9 10 11 12

18 19 20 21

13 22 23 2414 15 16

5 6 7 81098

171615

1 2 2 3 3 4

Figure 6. A medium network.

Table 4. Pipeline parameters of the network in Figure 7.

No. Length (m) Diameter (mm) Reliability

1 1500 400 0.98012 1200 300 0.69253 1000 250 0.84274 1100 400 0.96875 1100 300 0.79446 1100 250 0.95657 1100 300 0.90488 1500 300 0.78269 1200 350 0.607410 1000 250 0.928611 1000 300 0.777912 1000 250 0.846213 1000 300 0.916814 1000 350 0.895315 1500 250 0.670516 1200 300 0.762317 1000 250 0.603918 800 300 0.966819 800 350 0.764120 800 250 0.957521 800 300 0.623222 1500 300 0.741123 1200 250 0.925324 1000 350 0.9742

1

4 5 6 7

11 12 13 14

9 10 11 12

18 19 20 21

13 23 2414 15 16

5 6 7 8

1 2 2 3 3 4

Figure 7. The best solution of medium network when p0 is 0.7.

1

4 5 7

11 12 13 14

9 10 11 12

18 19 20 21

13 2322 14 2414 15 16

5 6 7 810

1 2 2 3 3 4

Figure 8. The best solution of medium network when p0 is 0.8.

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network that has quite large solution space, the searching

rate of SAGA can still reach about 70%. This verifies that

SAGA is an excellent algorithm for the topology

optimisation of lifeline networks. The searching rate of

GA ranges between 20% and 40% and that of SAA is a

little smaller, less than 20%. For ACA, the searching rate

is less than 10%. Moreover, the searching rate of PSA is

very low and it does not even find the optimal network

when the threshold reliabilities are 0.7 and 0.8.

4.3 Example 3

Figure 10 presents an actual gas network, located in

Shenyang City of China, with two sources (the bigger

nodes), 492 edges and 391 nodes. Considering the effect of

corrosion on the pipeline, the seismic reliability of the

original network subjected to an earthquake of intensity

VIII is very low. About 94% of nodal reliabilities are even

lower than 0.1. In order to increase the seismic capability

of the network, engineers suggested a retrofit scheme by

adding 254 pipelines to form the network G in Equation

(1). Based on the suggested network topology, the five

algorithms are used to optimise the network with the

threshold reliability of 0.8. Herein, all the parameters of

GA, SAA, SAGA and PSA are as same as those in

Example 1. In order to give a feasible solution, some of the

parameters of ACA are modified. In particular, the

iteration number is 225 and the number of ants takes the

value of 20. It is found that, in order to guarantee to obtain

a feasible network, many edges should be added to this

network, i.e. the probability that an edge selects 1 should

be much larger than that it selects 0. Therefore, all the

initial pheromones of selecting 0,ti;0ð1Þ, take the value of 2and those of selecting 1,ti;1ð1Þ, take the value of 20. In

addition, Q ¼ 1000 and r ¼ 0:2.However, as SAA cannot converge to a feasible

solution even after it is repeated 100 times, only the results

given by other four algorithms are reported. The four

11

9

2 2 43 3

5 6 87 10

15 10 16 1211 17

2213 14 23 1615 24

212018

14131211

7654

Figure 9. The best solution of medium network when p0 is 0.9.

Table 5. The best solutions for the network in Figure 7.

Reliabilityconstraint ( p0)

Lowest nodereliability Cost (RMB)

0.7 0.7023 20,740,0000.8 0.8008 22,370,0000.9 0.9110 26,020,000

Table 6. Searching rate for the network in Figure 7.

Algorithm P0 ¼ 0.7 (%) P0 ¼ 0.8 (%) P0 ¼ 0.9 (%)

GA 20 24 39SAA 18 19 4SAGA 69 72 79ACA 5 7 7PSA 0 0 2

Figure 10. An actual gas supply system.

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algorithms repeated 10 times and the best solutions

obtained are given in Table 7. Again, for this large

network, SAGA performs better than the other algorithms.

For the best solution given by SAGA, 69 pipelines can be

removed from the network and the resulting cost is

¥761,800,000. The optimal network given by SAGA is

shown in Figure 11. In contrast, for the best solution given

by GA, only 37 pipelines are removed and the cost is

¥790,420,000, i.e. ¥28,620,000 higher than the best

solution given by SAGA. Meanwhile, though ACA and

PSA give the feasible solutions, their performances are

still poor. For the best solutions given by ACA and PSA,

only 12 and 10 pipelines can be removed, respectively, and

the costs are much higher than the costs of the best

solutions given by SAGA and GA.

Apparently, it can be concluded that for the topology

design of lifeline networks subjected to earthquake, SAGA

is excellent, GA performs well, SAA performs moder-

ately, ACA follows and PSA performs rather poorly.

Actually, GA is a parallel algorithm as it consists of many

genes in one generation. Therefore, a lot of information

contained in the previous networks is used and preserved

during the selection, crossover and mutation operators.

Hence, GA can search the good results quickly and

efficiently. Conversely, SAA is a series algorithm, i.e. only

one network is checked during each iteration and thus little

information of previous networks is preserved. Although it

has been proven that SAA can reach the optimal solution if

the computational time is long enough (Xing & Xie,

2005), this is unacceptable in engineering practice.

For ACA, the new generated networks depend on the

value of pheromones. Although many networks are

generated to form a new group, it is still like a series

algorithm because they are all generated by the same

selection probability determined by the pheromones.

Meanwhile, for PSA, although it seems a parallel

algorithm, the new generated networks evolve from the

previous networks. However, the evolution process, in

fact, only uses the information of two best networks, one is

the current group and another is the previous evolution

steps. Therefore, it does not perform well. Moreover, it is

found that the effects of evolution are limited because each

bit can have only two values, 0 or 1; thus, it can only

evolve from 0 to 1 or from 1 to 0. Thus, although it is better

than ACA for the optimisation of water distribution

network (Liu et al., 2012a), it performs worse than ACA

for the optimisation model in this study. Finally, on the

basis of GA, the hybrid algorithm SAGA widens the

search space by SAO. It is therefore to be expected that

SAGA performs better than the other algorithms.

5. Concluding remarks

In this study, a comparative study has been performed

among five evolutionary optimisation algorithms, i.e. a

GA, a SAA, a SAGA an ACA and a PSA, through their

Table 7. Optimised solutions for the actual network in Figure11.

Algorithm Cost (RMB)Number of pipelines removed from

the proposed network

GA 790,420,000 37SAGA 761,800,000 69ACA 854,070,000 12PSA 866,630,000 10

Figure 11. An optimised solution gained using SAGA.

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performance investigation in topology optimisation of

lifeline networks. For this purpose, three modelled lifeline

networks including two benchmark networks and one

actual network have been presented. The concluding

remarks are as follows:

(1) As to the network with 10 nodes and 14 edges,

SAGA has an excellent performance that is able to

gain the best solution almost at each time. GA

performs well and its searching rate is larger than

50%. SAA follows and it achieves a searching rate

of 20–30%. Although ACA has a bad perform-

ance, it is better than PSA.

(2) As to the network with 16 nodes and 24 edges, the

performance of SAGA is still the best and it leads

to a searching rate larger than 70%. GA and SAA

perform inadequately for this size of networks.

The searching rate of GA is just over 20% while

that of SAA is below 20%. Meanwhile, ACA and

PSA have a worst performance: the searching rate

of ACA is less than 10%, while PSA has a

searching rate less than 2% and even cannot find

the optimal network in case that the threshold

values of reliability constraint are 0.7 and 0.8.

(3) As to the actual gas supply network, SAA fails,

while SAGA could still achieve a solution which is

better than those gained by other algorithms.

The presented hybrid scheme SAGA, therefore,

constitutes an excellent algorithm for seismic topology

optimisation of lifeline networks by replacing the mutation

operator in GA with the perturbations and updates used in

SAA.

Acknowledgements

The support from the Natural Science Funds of China (Grant No.51278380) and the National Key Technology R&D Program(Grant No. 2011BAK02B04) is greatly appreciated.

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